Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions

Jerome Droniou, Juan-Luis Vazquez

Research output: Contribution to journalArticleResearchpeer-review

18 Citations (Scopus)

Abstract

We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.
Original languageEnglish
Pages (from-to)413 - 434
Number of pages22
JournalCalculus of Variations and Partial Differential Equations
Volume34
Issue number4
DOIs
Publication statusPublished - 2009
Externally publishedYes

Cite this

@article{31cdf994f30446f78e3af8c6addd0a75,
title = "Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions",
abstract = "We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.",
author = "Jerome Droniou and Juan-Luis Vazquez",
year = "2009",
doi = "10.1007/s00526-008-0189-y",
language = "English",
volume = "34",
pages = "413 -- 434",
journal = "Calculus of Variations and Partial Differential Equations",
issn = "0944-2669",
publisher = "Springer",
number = "4",

}

Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions. / Droniou, Jerome; Vazquez, Juan-Luis.

In: Calculus of Variations and Partial Differential Equations, Vol. 34, No. 4, 2009, p. 413 - 434.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions

AU - Droniou, Jerome

AU - Vazquez, Juan-Luis

PY - 2009

Y1 - 2009

N2 - We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.

AB - We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.

UR - http://www.springerlink.com/content/71123gw40740j0q8/?MUD=MP

U2 - 10.1007/s00526-008-0189-y

DO - 10.1007/s00526-008-0189-y

M3 - Article

VL - 34

SP - 413

EP - 434

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 4

ER -